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Time-MultipatchDiscontinuous Galerkin

Space-Time IsogeometricAnalysis of ParabolicEvolution Problems

C. Hofer, U. Langer, M. Neumüller, I.Toulopoulos

RICAM-Report 2017-26

Time-Multipatch Discontinuous Galerkin Space-Time

Isogeometric Analysis of Parabolic Evolution Problems

Christoph Hofer1, Ulrich Langer1,2,3, Martin Neumuller2, and Ioannis Toulopoulos3

1 Doctoral Program “Computational Mathematics”Johannes Kepler University,

2 Institute of Computational Mathematics,Johannes Kepler University,

3 Johann Radon Institute for Computational and Applied Mathematics,Austrian Academy of Sciences

Altenbergerstr. 69, A-4040 Linz, Austria,

[email protected]@numa.uni-linz.ac.at

[email protected]@ricam.oeaw.ac.at

Abstract. In this paper, we present a new time-multipatch discontinuous Galerkin Isogeometric Analysis(IgA) technology for solving parabolic initial-boundary problems in space and time simultaneously. We provecoercivity of the discrete IgA problem with respect to a suitably chosen norm that together with boundedness,consistency and approximation results yields a priori discretization error estimates in this norm. Furthermore,we provide efficient parallel generation and parallel multigrid solution technologies, and present first numericalresults on massively parallel computers.

Key words: Parabolic initial-boundary value problems, Space-time Isogeometric Analysis, Time discontin-uous Galerkin methods, Space-time multigrid solvers.

1 Introduction

Fully discrete schemes for parabolic initial-boundary value problems (IBVP) are usually derivedby discretizing either first in space by means of some spatial discretization method like the fi-nite element method and then in time by some time-stepping method or vice versa. The formermethods are called vertical methods of lines [58], the latter horizontal methods of lines or Rothe’smethods [33]. Time-stepping methods are sequential in time. To overcome this curse of sequen-tiality on massively parallel computers, one needs some smart ideas for the parallelization intime. Time parallel methods have a long and exciting history that can be found in the very nicepaper [18] on 50 Years of Time Parallel Time Integration. Space-time finite element methods forparabolic and hyperbolic Partial Differential Equations (PDEs) go back to the 80s and 90s of thelast century [27, 28, 26, 3, 4, 20], and enjoy a real revival during the last couple of years due to theavailability of massively parallel computers with thousands or hundred thousands of cores, see,e.g., [49], [44], [1], [6], [43], [41], [60], [51], [2], [50], [5], [38], [59] for some resent mathematicalpapers related to parabolic problems. Moreover, there are several recent papers on the efficientuse of various space-time methods for solving exciting engineering problems involving movingcomputational spatial domains or / and interfaces see, e.g., [8], [56], [53], [54], [55], [57], [29], [30]and the references therein.

In [35], we were inspired by looking at the time variable t in a parabolic problem as just an-other variable, say, xd+1 if x1, . . . , xd are the spatial variable, and at the time derivative as a strongconvection in the direction xd+1 that can numerically be treated in a stable way by special dis-cretization techniques known from convection dominated elliptic convection-diffusion problems,see, e.g., [52]. The most popular stabilizing method is the Streamline-Upwind Petrov-Galerkin(SUPG) method introduced in [24]. We have used time-upwind test functions to construct stablesingle-patch space-time Isogeometric Analysis (IgA) schemes the discrete bilinear form of whichis coercive (elliptic) on the IgA space with respect to a suitably chosen mesh-dependent energy

2 C. Hofer, U. Langer, M. Neumuller, I. Toulopoulos

norm. This coercivity (ellipticity) property together with a corresponding boundedness property,consistency and approximation results for the IgA spaces yields the corresponding a priori dis-cretization error estimate. A posteriori error estimates that can be used for space-time adaptivityare derived in [34].

IgA was introduced in [25] as a new discretization methodology for PDE-based models. Thecore idea of IgA is to use the same smooth and high-order superior finite dimensional B-spline orNURBS spaces for parametrizing the computational domain and for approximating the solutionof the PDE model under consideration. IgA approaches have successfully been applied to thesolution of a wide range of linear and nonlinear problems. Their benefits have been highlighted inmany publications, see, e.g., the monograph [13], the survey paper [10] and the references therein.Although results related to the approximate properties of B-splines and their use for discretizingPDEs existed before, see, e.g., [48] and [23], the theoretical frame involving the parametrizationmappings has been started in [7], where the authors studied the approximation properties of B-splines (NURBS) in bent Sobolev spaces. In particular, they showed that the mapped B-splineshave the same approximation order in terms of the mesh size h as the piecewise polynomials ofthe same degree p, see also [9], [11], [10] for the generalization of these approximation results.

In this paper, we generalize the results of [35] from the single-patch to time-multipatch dis-continuous Galerkin (dG) space-time IgA schemes. As in [35], we consider the linear parabolicIBVP, find u : Q→ R such that

∂tu−∆u = f in Q, u = 0 on Σ, and u = u0 on Σ0, (1.1)

as a typical parabolic model problem posed in the space-time cylinder Q = Ω × [0, T ] = Q∪Σ ∪Σ0 ∪ ΣT , where ∂t denotes the partial time derivative, ∆ is the Laplace operator, f is a givensource function, u0 are the given initial data, T is the final time, Q = Ω×(0, T ), Σ = ∂Ω×(0, T ),Σ0 := Ω × 0, ΣT := Ω × T, and Ω ⊂ Rd (d = 1, 2, 3) denotes the spatial computationaldomain with the boundary ∂Ω. The spatial domain Ω is supposed to has a singlepatch NURBSrepresentation as is used in CAD [47]. More precisely, the space-time cylinder Q = ∪Nn=1Qn iscomposed of N subcylinders (patches or time slices) Qn = Ω × (tn−1, tn), n = 1, . . . , N , where0 = t0 < t1 < . . . < tN = T is some subdivision of time interval [0, T ]. The time faces between thetime patches are denoted by Σn = Qn+1∩Qn = Ω×tn, where ΣN = ΣT . Every space-time patch

Qn = Φn(Q) in the physical domain Q can be represented as the image of the parameter domain

Q = (0, 1)d+1 by means of a sufficiently regular IgA (B-Spline, NURBS etc.) map Φn : Q → Qn

that can be easily constructed from the spatial IgA map from Ω = (0, 1)d to Ω. In particular, eachQn can have its own mesh defined according to the characteristics of the problem. Therefore, theIgA spaces Vh0, which we are going to use, are smooth in each time patch Qn, but discontinuousacross the time faces Σn. For stabilizing the time discretization, the method incorporates ideas ofstreamline diffusion methodology. The continuity of the patch-wise defined approximate solutionsis ensured by introducing simple “up-wind” jump terms across the interfaces. The jump termsdo not include normal fluxes. This simplifies the error analysis. Moreover, the whole methodcan easily be materialized on a parallel platform. We develop a thoroughly theoretical study ofthe method. After defining the appropriate discontinuous B-spline spaces and after defining therelated discrete norm, we prove that the produced discrete bilinear form is coercive (elliptic) withrespect to this norm. This property ensures uniqueness and existence of the IgA solution. Basedon this ellipticity result, a related boundedness result and the consistency of the discrete bilinearform, we can easily estimate the discretization error by the best approximation error with respectto the discrete norm. With the help of the approximation results from [11] and [10], we derivediscretization error estimates taking into account that the exact solution can exhibit anisotropicregularity behavior, i.e., different regularity properties with respect to the time and to the spacedirections.

Finally, we have to solve one huge linear system Lhuh = fh of IgA equations defining allcontrol points in space and time all at once. The fast generation and the fast solution of this

Time dG space-time IgA of parabolic problems 3

system is an issue. In our case, we can benefit from the special time-multipatch dG structureof the discretization that leads to a block-bidiagonal system matrix Lh = blockbidiag(−Bi,Ai),where the block-diagonal matrices Ai, i = 1, . . . , N , and the block-subdiagonal matrices Bi,i = 2, . . . , N , have tensor product representations. These properties lead to a fast generation ofthe matrix Lh. The block-bidiagonal structure of the system matrix Lh enables us to solve thesystem sequentially from one space-time patch to the next space-time patch similar to a timestepping scheme. However, we want to overcome this curse of sequentiality since we want to usethe power of massively parallel computers with hundred or thousands of cores to solve this systemefficiently. Similar to [19], we propose a space-time multigrid method that solves the completesystem Lhuh = fh in parallel. In fact, we use the space-time multigrid method as preconditioner ina GMRES solver. The numerical results presented in the paper confirm not only our convergencerate estimates but also show the efficiency of the generation and solver technology proposed inthe paper. The first numerical results for the lowest-order splines can be found in our proceedingspaper [36] where we consider the simplified case of same smoothness of the solution in space andtime.

The remainder of the paper is organized as follows. In Section 2, beside introducing somenotations and preliminaries, the stable time-multipatch dG space-time IgA scheme is derived.Section 3 provides a complete a priori discretization error analysis, whereas Section 4 gives thematrix representation of our time-multipatch dG space-time IgA scheme and describes the par-allel space-time multigrid solver. Finally, we present and discuss some first numerical results inSection 5, and draw some conclusions in Section 6.

2 The model problem and its stable space-time IgA scheme

2.1 Preliminaries

Let Ω be a bounded Lipschitz domain in Rd, d = 1, 2, or 3, with the boundary Γ = ∂Ω. Forany multi-index α = (α1, . . . , αd) of non-negative integers α1, . . . , αd, we define the differentialoperator ∂αx = ∂α1

x1. . . ∂αd

xd, with ∂xj = ∂/∂xj, j = 1, . . . , d. For a non-negative integer `, C`(Ω)

denotes the space of all continuous functions v : Ω → R whose partial derivatives ∂αx v of allorders |α| =

∑dj=1 αj ≤ ` are continuous in Ω. As usual, L2(Ω) denotes the Lebesgue space for

which∫Ω|v|2 dx <∞, endowed with the norm ‖v‖L2(Ω) =

( ∫Ω|v(x)|2 dx

) 12, and L∞(Ω) denotes

the functions that are essentially bounded. We define the standard Sobolev space

H`(Ω) = v ∈ L2(Ω) : ∂αx v ∈ L2(Ω) for all |α| ≤ `,

endowed with the norm‖v‖H`(Ω) =

( ∑0≤|α|≤`

‖∂αx v‖2L2(Ω)

) 12 ,

whereas the trace space of H1(Ω) is denoted by H12 (Γ ). Further, we introduce the subspace

H10 (Ω) = v ∈ H1(Ω) : v = 0 onΓ of all functions v from H1(Ω) with zero traces on Γ . Let

J = (0, T ) with some final time T > 0 be the time interval. For later use, we define the space-timecylinder Q = Ω × J and its boundary parts Σ = ∂Ω × J , ΣT = Ω × T and Σ0 = Ω × 0such that ∂Q = Σ ∪Σ0 ∪ΣT , see an illustration in Fig. 1(a). Accordingly to the definition of ∂αx ,we now define the spatial gradient ∇xv = (∂x1v, . . . , ∂xdv). Let ` and m be positive integers. Forfunctions defined in the space-time cylinder Q, we define the Sobolev spaces

H`,m(Q) = v ∈ L2(Q) : ∂αx v ∈ L2(Q) for 0 ≤ |α| ≤ `, and ∂itv ∈ L2(Q), i = 1, ...,m, (2.1)

and, in particular, the subspaces

H1,00 (Q) =v ∈ L2(Q) : ∇xv ∈ [L2(Q)]d, v = 0 onΣ and (2.2)


H1,10,0

(Q) =v ∈ L2(Q) : ∇xv ∈ [L2(Q)]d, ∂tv ∈ L2(Q), v = 0 onΣ, v = 0 onΣT. (2.3)

We equip the above spaces with the norms and seminorms

‖v‖H`,m(Q) =( ∑|α|≤`

‖∂(α1,...,αd)x v‖2

L2(Q) +m∑

m0=0

‖∂m0t v‖2

L2(Q)

) 12 and (2.4a)

|v|H`,m(Q) =( ∑|α|=`

‖∂(α1,...,αd)x v‖2

L2(Q) + ‖∂mt v‖2L2(Q)

) 12 . (2.4b)

We recall Cauchy - Schwarz and Young’s inequalities∣∣∣∣∫Ω

uv dx

∣∣∣∣ ≤ ‖u‖L2(Ω)‖v‖L2(Ω) and

∣∣∣∣∫Ω

uv dx

∣∣∣∣ ≤ ε

2‖u‖2

L2(Ω) +1

2ε‖v‖2

L2(Ω) (2.5)

that hold for all functions u and v from L2(Ω) and for any fixed ε ∈ (0,∞). We also recallFriedrichs’ inequality that we later need in the form

‖v‖L2(Q) ≤ CΩ‖∇xv‖L2(Q), (2.6)

that holds for all v ∈ H1(Q) with vanishing trace on Σ, see proof of Friedrichs’ inequality in [12].

In what follows, positive constants c and C appearing in inequalities are generic constantswhich do not depend on the mesh-size h. In many cases, we will indicate on what may theconstants depend for an easier understanding of the proofs. Frequently, we will write a ∼ bmeaning that c a ≤ b ≤ C a with generic positive constants c and C.

2.2 The model parabolic problem

Using the standard procedure and integration by parts with respect to both x and t, we can easilyderive the following space-time variational formulation of (1.1): find u ∈ H1,0

0 (Q) such that

a(u, v) = l(v) for all v ∈ H1,10,0

(Q), (2.7)

with the bilinear form

a(u, v) = −∫Q

u(x, t)∂tv(x, t) dx dt+

∫Q

∇xu(x, t) · ∇xv(x, t) dx dt (2.8)

and the linear form

l(v) =

∫Q

f(x, t)v(x, t) dx dt+

∫Ω

u0(x)v(x, 0) dx, (2.9)

where the source f ∈ L2(Q) and the initial conditions u0 ∈ L2(Ω) are given.

For simplicity, we only consider homogeneous Dirichlet boundary conditions on Σ. However,the analysis presented in our paper can easily be generalized to other constellations of boundaryconditions. The space-time variational formulation (2.7) has a unique solution, see, e.g, [31] and[32]. In these monographs, beside existence and uniqueness results, one can also find useful apriori estimates and regularity results.

Assumption 2.1 We assume that the solution u of (2.7) belongs to V = H1,00 (Q)∩H`,m(Q) with

some ` ≥ 2 and m ≥ 1.


2.3 B-spline spaces and patch parametrizations

In this section, we briefly present the B-spline spaces and the form of the B-spline parametrizationsfor the physical space-time patches (subdomains). We refer to [13], [14] and [48] for a more detailedintroduction to B-splines.

To describe more clearly the basic materials, we start with presenting the B-spline spaces forthe univariate case. Let the integer p denotes the B-spline degree and the integer n1 denotes thenumber of the basis functions. Consider, Z = 0 = z1, z2, . . . , zM = 1 to be a partition of [0, 1]with [zj, zj+1], j = 1, . . . ,M − 1 to be the intervals of the partition. Based on Z, we considera knot-vector Ξ = 0 = ξ1 ≤ ξ2 ≤ . . . ≤ ξn1+p+1 = 1 and the associated vector of the knotrepetitions M = m1, . . . ,mM, this means,

Ξ = 0 = ξ1, . . . ξm1︸︷︷︸=z1

, ξm1+1 = . . . = ξm1+m2︸︷︷︸=z2

, . . . , ξn1+p+1−mM, . . . , ξn1+p+1 = 1︸︷︷︸=zM

. (2.10)

We assume that mj ≤ p for all internal knots. The B-spline basis functions are defined by theCox-de Boor formula

Bi,p =x− ξiξi+p − ξi

Bi,p−1(x) +ξi+p+1 − xξi+p+1 − ξi+1

Bi+1,p−1(x) (2.11)

where Bi,0(x) = 1 if ξi ≤ x ≤ ξi+1, and 0 otherwise.The multivariate B-spline spaces can be derived through tensor product procedures of the

univariate spaces. Let us consider the unit cube Q = (0, 1)d+1 ⊂ Rd+1, which we will refer to asthe parametric domain. Following the same steps, let the integers pk and nk, which denote thegiven B-spline degree and the number of basis functions of the B-spline space in xk-direction withk = 1, . . . , d + 1. We introduce the corresponding open-knot vectors Ξk

n = 0 = ξk1 ≤ ξk2 ≤ . . . ≤ξknk+p+1 = 1, the vectors Zk and Mk. We associate with each knot vector Ξk the B-spline basis

functions BΞk,pk of degree pk, for k = 1, . . . , d + 1. On the parametric domain Q, we define the

tensor-product B-spline space BΞd+1n ,p = ⊗d+1

k=1BΞkn,p

, where Ξd+1n = (Ξ1

n, . . . , Ξkn, . . . , Ξ

d+1n ),.

The decomposition into space-time patches In practice, it is usually more convenient todescribe the computational domain as a union of subdomains (patches) and to develop a multi-patch IgA approach. For our case, we will describe the space-time cylinder Q as a union of non-overlapping space-time patches Q1, Q2, . . . , QN . Consider a partition 0 = t0 < t1 < ... < tN = Tof [0, T ] and let Jn = (tn−1, tn). We define Qn = Ω × Jn and Σn = Qn+1 ∩Qn = Ω × tn wherewe identify ΣT and ΣN . In that way, we have

Q =N⋃n=1

Qn, with Qn+1 ∩Qn = Σn. (2.12)

A schematic illustration for the general case Q ⊂ Rd+1 is presented in Fig. 1(a), for the case ofQ ⊂ R2 in Fig. 1(b), and for Q ⊂ R3 in Fig. 1(c). We proceed below by defining the approximationB-spline spaces in every Qn as well the corresponding parametrizations.

Let us assume for simplicity that the B-spline degree is the same for all directions and forall the patches, i.e., pn,k = p for k = 1, . . . , d + 1, and let the integer nk denote the the numberof basis functions of the B-spline space in xk-direction, respectively. For every Qn,n = 1, . . . , N ,we introduce the (d + 1)−dimensional vector of knots Ξd+1

n = (Ξ1n, . . . , Ξ

kn, . . . , Ξ

d+1n ), with the

particular components given by Ξkn = 0 = ξk1 ≤ ξk2 ≤ . . . ≤ ξknk+p+1 = 1. For all the internal

knots, we assume that mkj ≤ p, with mk

j to be the associated multiplicities. The components Ξkn

of Ξd+1n form a mesh T

(n)

hn,Q= EmMn

m=1 in Q, where Em are the micro elements and hn is the mesh

size, which is defined as follows: given a micro element Em ∈ T(n)

hn,Q, we set hEm

= diam(Em),


and we define hn = maxhEm. We set h = max

n=1,...,Nhn. We refer the reader to [13] for more

information about the meaning of the knot vectors in CAD and IgA.Given the knot vector Ξk

n in every direction k = 1, . . . , d + 1, we construct the associated

univariate B-spline basis, BΞkn,p

= B(n)1,k (xk), . . . , B

(n)nk,k

(xk) using the Cox-de Boor recursion

formula (2.11), see, also [13] and [14] for more details. Accordingly, on the mesh T(n)

hn,Q, the basis

functions of the multivariate B-spline space BΞd+1n ,p are defined by the tensor-product of the

corresponding univariate B-spline basis functions of BΞkn,p

spaces, that is

BΞd+1n ,p = ⊗d+1

k=1BΞkn,p

= spanB(n)j (x)nB=n1·...·nk·...·nd+1

j=1 , (2.13)

where each B(n)j (x) has the form

B(n)j (x) =B

(n)j1

(x1) · . . . · B(n)jk

(xk) · . . . · B(n)jd

(xd), with B(n)jk

(xk) ∈ BΞkn,p. (2.14)

According to the IgA approach, every patch is described as a parametrization mapping by meansof the a B-spline space defined in the in the parametric domain and the control grid. Preciselyfor our case, we assume that we are given the net of the control points C

(n)j related to Qn, and

we parametrize each space-time patch Qn by

Φn : Q→ Qn, x = Φn(x) =

nB∑j=1

C(n)j B

(n)j (x) ∈ Qn, (2.15)

where x = Φ−1n (x), n = 1, . . . , N , cf. [13]. For every Qn, n = 1, . . . , N , we construct a mesh

T(n)hQn ,Qn

= EmMnm=1, where the elements Em are the images of Em ∈ T

(n)

hn,Qunder Φn, i.e.,

Em = Φn(Em). Also, for each E ∈ T(i)hi,Ωi

, we denote its support extension by E, where thesupport extension is defined to be the interior of the set formed by the union of the supportsof all B-spline functions whose supports intersects E. Accordingly to the parametric mesh, wedenote hEm = diam(Em) and define hQn = maxhEm : Em ∈ T (n)

hQn ,Qn, and the global physical

mesh size is h = maxhQn . For n = 1, . . . , N , we construct the B-spline space BΞd+1n ,p on Qn by

BΞd+1n ,p := spanB(n)

j |Qn : B(n)j (x) = B

(n)j Φ−1

n (x), for j = 1, . . . , nB. (2.16)

The global B-spline space Vh with components on every BΞdi ,p

is defined by

Vh := V(1)h1× . . .× V (N)

hN:= BΞd+1

1 ,p × . . .× BΞd+1N ,p. (2.17)

Assumption 2.2 The meshes T(n)

hn,Qare uniform, i.e., for every E ∈ T (n)

hn,Qthere exist a number

γn > 0 such that γn ≤ hn/ρE, where ρE is the radius of the inscribed circle of E.

Remark 2.1. Since the parametrizations Φn, n = 1, . . . , N , are fixed, under the Assumption 2.2,we have that hn ∼ hQn . Thus, below, we will use hn, for denoting any of the mesh sizes, parametricor physical. For simplicity, we assume that hn ≤ 1 for all n = 1, . . . , N .

The parametrization mappings Φn, n = 1, . . . , N can be considered to be bi-Lipschitz homeomor-phisms, [10]. For simplifying the analysis, we further consider the following regularity propertieson Φn, n = 1, . . . , N .

Assumption 2.3 Assume that every Φn and Φ−1n , n = 1, ..., N , are sufficiently smooth, (C1

diffeomorphisms), and there exist constants 0 < c < C such that c ≤ | det JΦn| ≤ C, where JΦn

is the Jacobian matrix of Φn, i.e., JΦn =∂(Φn,1, . . . ,Φn,d+1)

∂(x1, . . . , t).


Assumption 2.3 helps on simplifying the form of the constants, which appear in the relationsbetween the norms of the pull-back solution and the physical relevant solution.

Corollary 2.1. Let the Assumption 2.3 and let u ∈ H`,m(Q) with ` ≥ 2 and m ≥ 1. Then its

pull-back u = u Φn ∈ H1,1(Q) and there exist constants c1 and c2 depending only on Φn andΦ−1n and not on u, such that c1‖u‖H1,1(Q) ≤ ‖u‖H1,1(Qn) ≤ c2‖u‖H1,1(Q).

Σ0

ΣT

Rd

Ω

Σ

t

T

Ξnk=1Ξ

nk=2

Ξnk=3

Q

Qn

Φn

Space time Cylinder Q

(a)

T

t

u(x,t)

tn

tn+1

ΩΣ

n Σn+1

u(x,0)

Qn+1

X

(b)

Ω

T

t

tn

tn+1

u(x,y,t)

Σn

Σn+1

Row 1 Row 2 Row 3 Row 40

2

4

6

8

10

12

Column 1

Column 2

Column 3

(0,0,0

Qn+1

(c)

Fig. 1. (a) The decomposition of the space type cylinder Q into space-time patches Qn, together with the mesh in the

parametric domain Q produced by the knot-vectors Ξkn, (b) The space-time patches Qn with their interfaces and the graph

of u(x, t) for the case of Q ⊂ R2, (c) The space-time patches Qn with their interfaces and the graph of u(x, t) for the caseof Q ⊂ R3.

To keep the notation simple, in what follows, we will use the notation TN(Q) := Q1, Q2, . . . , QNfor the decomposition (2.12) and the sup-index n to denote the restrictions to Qn, e.g., un := u|Qn .We denote the global discontinuous B-spline space and the local continuous patch-wise B-splinespaces by

V0h =vh ∈ L2(Q) : vh|Qn ∈ BΞd+1i ,p(Qn), forn = 1, . . . , N, and vh|Σ = 0, (2.18a)

V(n)

0h =vh ∈ BΞd+1i ,p(Qn), forn = 1, . . . , N, and vh|Σ = 0. (2.18b)

Notice that vh ∈ V0h is discontinuous across Σn. We introduce the notation

vnh,+ = limε→0+

vh(tn + ε), vnh,− = limε→0−

vh(tn + ε), JvhKn = vnh,+ − vnh,−, JvhK0 = v0h,+, (2.19)

where JvhKn denotes the jump of vh across Σn for n ≥ 1, and JvhK0 = v0h,+ denotes the jump across

Σ0. Decomposition (2.12) helps us to consider N local problems posed on each space-time patchQn. In view of (2.19), for a smooth function u we have that JuKn = un+ − un− = 0 forn ≥ 1, andaccordingly define JuK0 = u|Σ0 .

2.4 Stable multipatch space-time dG IgA discretization

Let us consider the space-time patchQn, with the outer normal to ∂Q to be n = (n1, . . . , nd, nd+1) =

(nx, nt). For the being time, we assume that un−1 is known. Let vnh ∈ V(n)

0h and wnh = vnh+θn hn∂tvnh

with some positive parameter θn, that will be defined later. Note that wnh∣∣Σ

= 0. Multiplying


∂tu − ∆u = f by wnh , integrating over Qn, and applying integration by parts, we arrive at thevariational identity

aQn(u, vnh) =

∫Qn

∂t u (vnh + θn hn∂tvnh) +∇x u · ∇x v

nh + θn hn∇xu · ∇x∂tv

nh dx dt (2.20)

−∫∂Qn

nx · ∇xu(vnh + θn hn∂tvnh) ds+

∫Σn−1

un−1+ vn−1

h,+ ds

=

∫Qn

f (vnh + θn hn∂tvnh) dx dt+

∫Σn−1

un−1− vn−1

h,+ ds, for 1 ≤ n ≤ N,

where we used that un−1− = un−1

+ = un−1 on every Σn. Furthermore, using that nx|Σn = 0, andwh = 0 on Σ, we have

aQn(u, vnh) =

∫Qn

∂t u (vnh + θn hn∂tvnh) +∇x u · ∇x v

nh + θn hn∇xu · ∇x∂tv

nh dx dt (2.21a)

+

∫Σn−1

JuKn−1 vn−1h,+ ds =

∫Qn

f (vnh + θn hn∂tvnh) dx dt,

for all n = 1, . . . , N . Summing over all Qn, we conclude that

a(u, vh) =∑N

n=1 aQn(u, vnh) =∑N

n=1

∫Qnf (vnh + θn hn∂tv

nh) dx dt, (2.21b)

where above the a(·, ·) is considered to be the global bilinear form. Now, the space-time dG IgAvariational scheme for (1.1) can be formulated as follows: Find uh ∈ V0h such that

ah(uh, vh) = lh(vh), ∀vh ∈ V0h, (2.22a)

where

ah(uh, vh) =N∑n=1

aQn(uh, wnh)

=N∑n=1

∫Qn

∂t unh (vnh + θn hn∂tv

nh) +∇x u

nh · ∇x v

nh + θn hn∇xu

nh · ∇x∂tv

nh dx dt

+N∑n=2

∫Σn−1

JuhKn−1 vn−1h,+ ds+

∫Σ0

JuhK0v0h,+ ds,

lh(vh) =N∑n=1

∫Qn

f (vnh + θn hn∂tvnh) dx dt+

∫Σ0

u0 v0h,+ ds,

(2.22b)

where J·K is as in (2.19). Below for simplifying the jump expressions, we use the consolidatedexpression

∑Nn=1

∫Σn−1

JuhKn−1 vn−1h,+ ds.

The classical properties for the discrete bilinear form. We cite a few auxiliary results thatwill be used in the error analysis below. For the proofs, we refer to [7], [15], [9], see also discussionin [37].

Lemma 2.1. Let the patch Qn ∈ TN(Q), and let v ∈ H1(Qn), vh ∈ BΞd+1n ,p and E ∈ T

(n)hn,Qn

.Then there are positive constants Ctr, Cinv,0 and Cinv,1 depending on Φn and the quasi-uniform

properties of T(n)hn,Qn

, such that

‖v‖2L2(∂E) ≤Ctrh−1

n

(‖v‖L2(E) + hn |v|H1(E)

)2, (2.23a)

‖vh‖2L2(∂E) ≤Cinv,0h−1

n ‖vh‖2L2(E), (2.23b)

‖∇vh‖2L2(E) ≤Cinv,1h−2

n ‖vh‖2L2(E). (2.23c)


By the inequalities (2.23), we can easily infer that

‖∂tvh‖2L2(∂E) ≤ Cinv,1h

−2n ‖vh‖2

L2(E), and ‖∂t∂xivh‖2L2(∂E) ≤ Cinv,1h

−2n ‖∂xivh‖2

L2(E). (2.24)

Motivated by (2.22b), we define the norm on V0h

‖v‖dG =( N∑n=1

(‖∇xv‖2

L2(Qn) + θn hn ‖∂tv‖2L2(Qn) +

1

2‖JvKn−1‖2

L2(Σn−1)

)+

1

2‖v‖2

L2(ΣN )

) 12. (2.25)

Lemma 2.2. The discrete bilinear form ah(·, ·) defined in (2.22b) is V0h-elliptic, i.e., holds

ah(vh, vh) ≥ Ce‖vh‖2dG, for vh ∈ V0h, (2.26)

where Ce = 12

for θn ≤ C−2inv,0.

Proof. Using Green’s formula

∫Qn

∂tvh vh + vh ∂tvh dx dt =

∫∂Qn

ntv2h ds, we obtain the identity

∫Qn

∂tvh vh =1

2

∫Qn

∂t v2h dx dt =

1

2

∫Σn

(vh,−)2 ds− 1

2

∫Σn−1

(vh,+)2 ds. (2.27)

The definition of aQn and identity (2.27) yield

aQn(vh, vh) =

∫Qn

1

2∂tv

2h + θnhn(∂tvh)

2 + |∇xvh|2 +θnhn

2∂t|∇xvh|2 dx dt+

∫Σn−1

JvhKn−1vn−1h,+ ds

=

∫Qn

θnhn(∂tvh)2 + |∇xvh|2 dx dt+

∫∂Qn

θnhn2|∇xvh|2nt

+

∫Σn−1

((vn−1h,+ )2 − vn−1

h,− vn−1h,+ −

1

2(vn−1h,+ )2

)ds+

1

2

∫Σn

(vnh,−)2 ds

=θnhn‖∂tvh‖2L2(Qn) + ‖∇xvh‖2

L2(Qn) +θnhn

2

(‖∇xvh‖2

L2(Σn) − ‖∇xvh‖2L2(Σn−1)

)+

∫Σn−1

((vn−1h,+ )2 − vn−1

h,− vn−1h,+ −

1

2(vn−1h,+ )2

)ds+

1

2

∫Σn

(vnh,−)2 ds

≥θnhn‖∂tvh‖2L2(Qn) + ‖∇xvh‖2

L2(Qn) −θnhn

2‖∇xvh‖2

L2(Σn−1)

+

∫Σn−1

(1

2(vn−1h,+ )2 − vn−1

h,− vn−1h,+

)ds+

1

2

∫Σn

(vnh,−)2 ds

≥θnhn‖∂tvh‖2L2(Qn) +

(1−

θnC2inv,0

2

)‖∇xvh‖2

L2(Qn)

+

∫Σn−1

(1

2(vn−1h,+ )2 − vn−1

h,− vn−1h,+

)ds+

1

2

∫Σn

(vnh,−)2 ds, (2.28)

where we have used (2.23) at the last step in (2.28). Summing over all Qn, and using (2.19), weobtain

ah(vh, vh) =N∑n=1

aQn(vh, vh) ≥N∑n=1

θnhn‖∂tvh‖2L2(Qn) +

(1−

θnC2inv,0

2

)‖∇xvh‖2

L2(Qn)

+N∑n=1

1

2‖JvhKn−1‖2

L2(Σn−1) +1

2‖vNh,−‖2

L2(ΣN ).

Choosing 0 < θn ≤ C−2inv,0 the result follows easily by setting Ce = 1

2.


Remark 2.2 (a-priori bound). Using the discrete solution uh as test function in (2.22a), the in-equalities defined in (2.5) and (2.6) yield

Ce‖uh‖2dG ≤ ah(uh, uh) ≤

∣∣lh(uh)∣∣ ≤ ∣∣∣ N∑n=1

∫Qn

f (unh + θnhn∂tunh) dx dt

∣∣∣+∣∣∣ ∫

Σ0

u0 u0h,+ ds

∣∣∣≤‖f‖L2(Q)

( N∑n=1

‖unh‖2L2(Qn)

) 12 +

N∑n=1

(θnhn)12‖f‖L2(Qn) (θnhn)

12‖∂tunh‖L2(Qn) + ‖u0‖L2(Σ0)‖u0

h,+‖L2(Σ0)

≤‖f‖L2(Q)

( N∑n=1

CΩ‖∇xunh‖2

L2(Qn)

) 12

+ θmaxh‖f‖L2(Q)

( N∑n=1

θnhn‖∂tunh‖2L2(Qn)

) 12

+√

2‖u0‖L2(Σ0)‖uh‖DG

≤√

2cstab

(‖f‖L2(Q) + ‖u0‖L2(Σ0)

)‖uh‖dG,

(2.29)

where h = maxnhn, and cstabl depends on the constants in (2.6) and on θmax = maxnθn. By(2.29), we can immediately get the a-priori bound ‖uh‖dG ≤ C

(‖f‖L2(Q) + ‖u0‖L2(Σ0)

).

Later, in the discretization error analysis, we need continuity properties for ah(·, ·). Let V andV0h are the spaces defined in Assumption 2.1 and in (2.18). We define the space V0h,∗ = V + V0h

endowed with the norm

‖v‖dG,∗ =(‖v‖2

dG +N∑n=1

(θnhn)−1‖v‖2L2(Qn) +

N∑n=2

‖vn−1− ‖2

L2(Σn−1)

) 12. (2.30)

Lemma 2.3. Let u ∈ V0h,∗. Then for vh ∈ V0h holds

ah(u, vh) ≤ cb‖u‖dG,∗‖vh‖dG, (2.31)

where cb = max(Cinv,1 θmax, 2) with θmax = maxnθn ≤ C−2inv,0.

Proof. We recall (2.19) and JuK0 = u|Σ0 . For the first and the interface jump terms of ah, we use(2.27) and (2.5) and obtain

N∑n=1

(∫Qn

∂tu vh dx dt+

∫Σn−1

JuKn−1 vn−1h,+ ds

)=−

N∑n=1

∫Qn

u ∂tvh dx dt+N∑n=1

(∫Σn

u vh ds−∫Σn−1

u vh ds+

∫Σn−1

JuKn−1 vn−1h,+ ds

)≤( N∑n=1

(θnhn)−1(∫

Qn

u2 dx dt)2) 1

2( N∑n=1

θnhn

(∫Qn

∂tv2h dx dt

)2) 12

+N∑n=2

∫Σn−1

(vn−1h,− − v

n−1h,+ )un−1

− ds+

∫ΣN

vnh,−un− ds

≤( N∑n=1

(θnhn)−1(∫

Qn

u2 dx dt)2) 1

2( N∑n=1

θnhn

(∫Qn

∂tv2h dx dt

)2) 12

+( N∑n=2

‖vn−1h,− − v

n−1h,+ ‖

2L2(Σn−1)

) 12( N∑n=2

‖un−1− ‖2

L2(Σn−1)

) 12

+ ‖uN−‖L2(ΣN )‖vNh,−‖L2(ΣN )

≤( N∑n=1

(θnhn)−1(∫

Qn

u2 dx dt)2) 1

2( N∑n=1

θnhn

(∫Qn

∂tv2h dx dt

)2) 12

+√

2(1

2

N∑n=1

‖JvhKn−1‖2L2(Σn−1) +

1

2‖vNh ‖2

L2(ΣN )

) 12√

2( N∑n=2

‖un−1− ‖2

L2(Σn−1) +1

2‖uN−‖2

L2(ΣN )

) 12.

(2.32)


For the second term, an application of Cauchy-Schwartz yields

N∑n=1

∫Qn

(θnhn)12∂tu (θnhn)

12∂tvh dx dt+

N∑n=1

∫Qn

∇xu · ∇xvh dx dt (2.33)

≤( N∑n=1

θnhn‖∂tu‖2L2(Qn)

) 12( N∑n=1

θnhn‖∂tvh‖2L2(Qn)

) 12

+( N∑n=1

‖∇xu‖2L2(Qn)

) 12( N∑n=1

‖∇xvh‖2L2(Qn)

) 12.

For the last term, we apply Cauchy-Schwartz and inverse inequalities to show

N∑n=1

∫Qn

∇xu · (θnhn)∇x∂tvh dx dt ≤( N∑n=1

‖∇xu‖2L2(Qn)

) 12( N∑n=1

(θnhn)2

d∑i=1

∫Qn

(∂t∂xivh)2 dx dt

) 12

≤( N∑n=1

‖∇xu‖2L2(Qn)

) 12( N∑n=1

(θnhn)2Cinv,1h−2n

d∑i=1

∫Qn

(∂xivh)2 dx dt

) 12

(2.34)

≤Cinv,1θmax( N∑n=1

‖∇xu‖2L2(Qn)

) 12( N∑n=1

‖∇xvh‖L2(Qn)

) 12,

where θmax = maxnθn ≤ C−2inv,0. Gathering together the bounds (2.32), (2.33) and (2.34) and

setting cb = max(Cinv,1 θmax, 2) yields the desired result.

Lemma 2.4. Let Assumption 2.1 and let uh ∈ V0h be the dG IgA solution of (2.22a). Then

ah(u− uh, vh) = 0, for vh ∈ V0h. (2.35)

Proof. Let vh ∈ V0h. Since u ∈ V and wh = (vh + θnhn∂tvh)|Σ = 0 then∫Σn−1

JunKvh ds = 0, for

2 ≤ n ≤ N . Moreover, u satisfies (2.21b). Comparing (2.21) and (2.22), we can infer (2.35).

3 A priori discretization error analysis

Based on the quasi-interpolation estimates presented in [7],[10], see also [48] and [37], we construct

below quasi-interpolants Π(n)h : H`,m(Qn)→ BΞd+1

i ,p(Qn), for n = 1, . . . , N , suitable for providing

anisotropic interpolation estimates. Utilizing these estimates, we show the desirable anisotropicerror estimates at the end of this section.

3.1 Multivariate quasi interpolants in Q

Let Z = 0 = z1, z2, . . . , zM = 1 be a partition of I = (0, 1) with Ij = (zj, zj+1), j = 1, . . . ,M−1to be the intervals of the partition and with mesh size h = maxj|Ij|. Based on Z, we considera knot-vector Ξ = 0 = ξ1 ≤ ξ2 ≤ . . . ≤ ξn+p+1 = 1 and the associated vector of the knotmultiplicities M = m1, . . . ,mM. Let the integers s, ` be such that 0 ≤ s ≤ ` ≤ p + 1 and letf ∈ H`(I). Based on the quasi-interpolation estimates presented in [48], [10], we can construct a

quasi-interpolant Πh : H`(I)→ BΞ,p(I), such that the following interpolation estimate holds true

|f − Πhf |Hs(I) ≤ Ch`−s ‖f‖H`(I), (3.1)

where the constant C > 0 depends on p and uniformity parameters of the partition. The previ-ous construction of the univariate quasi-interpolation can be extended to the multi-dimensionalcase by applying tensor-product construction procedures as those presented in Section 2.3. Forexample, let f ∈ H`(Q) with ` ≥ 1, and for k = 1, . . . , d + 1, let ΠΞk

nbe the corresponding k-th


univariate quasi-interpolant onto BΞkn. We construct the multi-dimensional B-spline interpolant

ΠΞd+1n

as

ΠΞd+1nf = ⊗d+1

k=1ΠΞknf. (3.2)

The general quasi-interpolation properties of the produced multivariate B-spline interpolants areinherited by the corresponding properties of the univariate interpolants. We refer to [48], [7] and[10], for a comprehensive analysis for constructing tensor-product B-spline interpolants.

3.2 Anisotropic quasi-interpolation in space-time patches Qn

Let f ∈ H`,m(Q), with ` ≥ 2 and m ≥ 1, and as usual we denote by fn = f |Qn , for n = 1, . . . , N its

restriction on the space-time patches. Further, we denote by fn = f Φn its pull-back function. Wenote that fn in general does not inherit the regularity of f but rather belongs to a bent-Sobolevspace

H`(Q) = H`1(I)⊗ . . .⊗H`d(I)⊗H`d+1(I), (3.3)

that allows less regularity across the microelement interfaces, where in (3.3),H`i(I), i = 1, . . . , d+1 are the corresponding univariate bent-Sobolev spaces, [7, 10]. For showing the anisotropic quasi-interpolation estimates of interest, we have been strongly inspired by the results presented in [11]and in [10], which are suitable for anisotropic meshes. The generalization here is that we giveanisotropic interpolation estimates that follow the anisotropic regularity of the solution. In thespirit of (3.2), we define in Q, the interpolant ΠΞd+1

nfn of the pull-back fn. For simplicity, we

shall write Π(n)h instead of ΠΞd+1

n. Accordingly, we define in the space-time patches Qn the quasi-

interpolant of fn as Π(n)h fn =

(Π

(n)h fn

)Φ−1

n . By extension, we can define the global interpolant

Πh : H`,m(Q) → Vh as, (Πhf)|Qn = Π(n)h fn, see e.g., [10]. Before giving estimates on how well

Π(n)h f approximates f ∈ H`,m(Q), some terminology is required.

We recall by Section 2.1 the definition of the differential operator D(α,m), that is,

D(α,m)f := D(α1,...,αd,m)f =∂α1 . . . ∂αd∂m

∂xα11 . . . ∂xαd

d ∂tmf. (3.4)

In order to derive the anisotropic estimate, we need to introduce the derivatives with respect tothe coordinate system that is naturally introduced by the mappings Φn : Q → Qn, see (2.15).We note again that, the mappings Φn are constructed on relatively coarse meshes and are highlysmooth (polynomials) on the microelements of those meshes.

We recall that the columns of the Jacobian matrix of Φn, see (2.15) and Assumption 2.3, havethe form [∂Φn,1

∂xi, . . . ,

∂Φn,d∂xi

,∂Φn,d+1

∂xi

]>=[∂Φn,1∂xi

, . . . ,∂Φn,d∂xi

, 0]>, (3.5)

where we have used above that∂Φn,d+1

∂xi= 0 for i = 1, . . . , d, which easily follows by the tensor-

product constructing properties of each Φn. Denote gn,i(x, t) =[∂Φn,1

∂xi(Φ−1

n (x, t)), . . . ,∂Φn,d

∂xi(Φ−1

n (x, t)), 0].

We introduce the derivatives of f with respect to the spatial Φn coordinates. The first derivativesare just the directional derivatives with respect to gn,i for i = 1, . . . , d, i.e.,

∂f(x, t)

∂gn,1=∇f(x, t) · gn,1(x, t),

...

∂f(x, t)

∂gn,d=∇f(x, t) · gn,d(x, t).

(3.6a)


The “one-directional” high-order derivatives are defined accordingly as

∂αif

∂gαin,i

=∂

∂gn,i

(...( ∂f

∂gn,i

))︸︷︷︸

αi−times

. (3.6b)

Let the multi-index α = (α1, . . . , αd) be defined as in Section 2.1. In dealing with multi-directionderivatives, we introduce the notation

xα,q =xα11 , . . . , x

αdd , x

qd+1, with x ∈ Rd+1, q ∈ N0, (3.7a)

DαΦnf =

∂α1

∂gα1n,1

. . .∂αdf

∂gαdn,d

, Dα,qΦnf =

∂α1

∂gα1n,1

. . .∂αdf

∂gαdn,d

∂αqf

∂gqn,d+1

. (3.7b)

In relation to the Dα,qΦnf derivatives, we define the norms and seminorms

‖f‖2Hα,q

Φn(Qn) =

α1∑s1=0

. . .

αd∑sd=0

q∑sd+1=0

|f |2Hα,qΦn

(Qn),

|f |2Hα,qΦn

(Qn) =∑

E∈T (n)hn,Qn

|f |2Hα,qΦn

(E),(3.8)

where

|f |2Hα,qΦn

(E) =‖Dα,qΦnf‖L2(E).

We introduce the space Hα,qΦn

(Qn) endowed with the norm ‖ · ‖Hα,qΦn

(Qn) = ‖ · ‖Hα,qΦn

(Qn). Below, we

show the relation between the gn,i directional derivative norms and norms of the usual partialderivatives.

Proposition 3.1. Let f : Q → R be a smooth function and let the Assumption 2.3 and themulti-index α such that |α| = 1. For all E ∈ T (n)

hn,Qnwe have the relations

‖Dαf‖L2(E) ∼∑|α|=1 ‖D

α,0Φnf‖L2(E), (3.9a)

‖f‖H`,m(E) ∼∑|α|=1‖f‖H`α,m

Φn(E), (3.9b)

where the associated constants depend on p, γ, gn,i and Φn.

Proof. The inequalities (3.9) follow by definition (3.8) and (3.6).

Assumption 3.1 For simplicity, we assume that p+ 1 ≥ max(`,m), cf. Assumption 2.1.

Theorem 3.1. Let the multi-index α such that |α| = 1. Let the Assumptions 2.2, 2.3 and 3.1

hold and let E ∈ T (n)hn,Qn

and E be its support extension. Furthermore, let f ∈ H`,m(Qn) with ` ≥ 2and m ≥ 1. Then, the quasi-interpolation estimates(∑

E∈T (n)hn,Qn

|∇x(f −Π(n)h f)|2L2(E)

) 12 ≤Cx

(h`−1n + hmn

)‖f‖H`,m(Qn), (3.10a)(∑

E∈T (n)hn,Qn

|∂t(f −Π(n)h f)|2L2(E)

) 12 ≤Ct

(h`n + hm−1

n

)‖f‖H`,m(Qn), (3.10b)(∑

E∈T (n)hn,Qn

|f −Π(n)h f |2L2(E)

) 12 ≤C0

(h`n + hmn

)‖f‖H`,m(Qn), (3.10c)

holds, with Cx, Ct and C0 depending only on d, p, γ,gn,i and Φn.


Proof. We noted above that the last component of gn,i is equal to zero. This implies that the

derivatives ∂f(x,t)∂gn,i

, see (3.6), do not include terms like ∂f∂t

. Since f ∈ H`,m(Q), we have that

f ∈ H`α,0Φn

(Qn) ∩ H0,mΦn

(Qn), where α is a multi-index with d-components such that |α| = 1.Making use of the interpolation results presented in [10], see Theorem 4.18, we have

‖Dα,0Φn

(f −Π(n)h f)‖L2(E) ≤ c1

(h`−1n + hmn

)∑|α|=1‖f‖H`α,m

Φn(E). (3.11)

Using (3.9) and (3.11), we can derive the following interpolation estimate

‖∇x(f −Π(n)h f)‖L2(E) ≤ c2

∑|α|=1‖D

α,0Φn

(f −Π(n)h f)‖L2(E) ≤ c3

(h`−1n + hmn

)‖f‖H`,m(E), (3.12)

with c2 and c3 depending on d, p, γ,Φn. In (3.12) summing over all E ∈ T (n)hn,Qn

, we have that∑E

‖∇x(f −Π(n)h f)‖2

L2(E) ≤c4

(h`−1n + hmn

)2 ∑E

‖f‖H`,m(E),

≤c5

(h`−1n + hmn

)2 ∑E

∑E′∈E

‖f‖H`,m(E′) (3.13)

where c4 and c5 depend on the constant c3. Now, we observe that the last double sum in (3.13) con-

sists of repeated element norm terms as ‖f‖H`,m(E). More precisely, for every element E ∈ T (n)hn,Qn

,the related norm term ‖f‖H`,m(E) appears as many times in (3.13) as the number of the extension

supports E, lets say ENb,E, where the element E belongs. By the constructing nature of B-splines,ENb,E depends on the underlying B-spline degree and the knot repetitions mi, i.e., the smoothnessof B-splines across the microelement interfaces. Setting Emax,E = max

E∈T (n)hn,Qn

ENb,E, inequality

(3.13) gives

∑E

‖∇x(f −Π(n)h f)‖2

L2(E) ≤c5Emax,E

(h`−1n + hmn

)2

‖f‖H`,m(E) (3.14)

≤C2x

(h`−1n + hmn

)2

‖f‖H`,m(Qn),

and estimate (3.10a) follows. Following similar procedure, we can show the estimates (3.10b) and(3.10c).

Proposition 3.2. Let ` ≥ 2 and m ≥ 1 be integers and let f ∈ H`,m(Q). Let the Assumptions2.2, 2.3 and 3.1 hold, furthermore let Πn

hf be the corresponding quasi-interpolant defined above.Then there exist constants C∗i , i = 1, 2 independent of f and h but dependent on the constants of(2.23) and on (3.10) such that

‖f −Π(n)h f‖L2(∂Qn) ≤ C∗1

(h`− 1

2n + h

m− 12

n

)‖f‖H`,m(Qn), (3.15a)

‖f −Πhf‖dG,∗ ≤ C∗2(h`−1 + hm−

12

)‖f‖H`,m(Q). (3.15b)

Proof. Using the trace inequalities given in (2.23) and then (3.10), we have that

‖f −Π(n)h f‖2

L2(∂Qn) ≤ Ctr

(h−1n

(‖f −Π(n)

h f‖2L2(Qn) + hn ‖∇f −∇Π(n)

h f‖2L2(Qn)

))≤ C∗1 (h

2(`− 12

)n + h

2(m− 12

)n )‖f‖2

H`,m(Qn). (3.16)

Recalling the definition of ‖ · ‖dG,∗ and using again the estimates (3.10) and (3.15a), we obtain


‖f −Πhf‖2dG,∗ =

( N∑n=1

(‖∇xf −Π(n)

h f‖2L2(Qn) + θ h ‖∂tf −Π(n)

h f‖2L2(Qn)+

1

2‖J(f −Πhf)n−1K‖2

L2(Σn−1)

)+

1

2‖f −Π(N)

h f‖2L2(ΣN )

)+

N∑n=1

(θnh)−1‖f −Π(n)h f‖2

L2(Qn) +N∑n=2

‖(f −Π(n−1)h f)n−1

− ‖2L2(Σn−1)

≤N∑n=1

(C0,n (h2(`−1)

n + h2(m− 1

2)

n ) +C1,n (h2(`− 1

2)

n + h2(m− 1

2)

n ) +C2,n(θn hn)−1(h2`n + h2m

n ))‖f‖2

H`,m(Qn)

≤N∑n=1

((C0,n + C1,n + C2,n)

(h2(`−1)n + θnh

2`−1n + h2m−1

n + θnh2m−1n

)‖f‖2

H`,m(Qn)

≤ C∗2(h2(`−1) + h2(m− 12

)) ‖f‖2H`,m(Q), (3.17)

where a reduction of the terms h`−1 and h`−12 have been performed. This completes the proof of

(3.15b).

Theorem 3.2. Let u and uh solve (2.7) and (2.22a), respectively. and uh solve (2.22a).Under Assumption 2.1, there exist a c > 0, independent of h such that

‖u− uh‖dG ≤ c(h`−1 + hm−12 ) ‖u‖H`,m(Ω), (3.18)

Moreover, if 1 ≤ m < ` ≤ p+ 1

‖u− uh‖dG ≤ chm−12‖u‖2

H`,m(Ω). (3.19)

Proof. Using the properties of bilinear form ah(·, ·), i.e., V0h ellipticity and the boundedness, aswell as the consistency (2.35), we can obtain

‖uh−Πhu‖2dG ≤ c1ah(uh−Πhu, uh−Πhu) = ah(u−Πhu, uh−Πhu) ≤ c2‖u−Πhu‖dG,∗‖uh−Πhu‖dG.

(3.20)

Hence, applying triangle inequality ‖u− uh‖dG ≤ ‖u−Πhu‖dG,∗ + ‖uh −Πhu‖dG, we can derive

‖u− uh‖dG ≤ c‖u−Πhu‖dG,∗. (3.21)

Utilizing the estimate (3.15) in (3.21) yields (3.18). Estimate (3.19) is a direct result of (3.18).

Remark 3.1. We remark that for the case of highly smooth solutions, i.e., p+ 1 ≤ min(`,m), theestimate (3.18) takes the form

‖u− uh‖dG ≤ c hp ‖u‖H`,m(Ω). (3.22)

4 Matrix representation and space-time multigrid

We recall the discrete variational problem given in (2.22a), where we want to find uh ∈ V0h suchthat

ah(uh, vh) = lh(vh), ∀vh ∈ V0h,

with V0h := V(1)h × . . .× V (N)

h and

ah(uh, vh) =N∑n=1

aQn(unh, vnh),


where the local bilinear form for each space-time patch n = 1, . . . , N is given by

aQn(unh, vnh) =

∫Qn


nh) +∇x u

nh · ∇x(v

nh + θn hn∂tv

nh) dx dt+

∫Σn−1

JuhKn−1 vn−1h,+ ds

=

∫Qn


nh) +∇x u

nh · ∇x(v

nh + θn hn∂tv

nh) dx dt+

∫Σn−1

un−1h,+ vn−1

h,+ ds

−∫Σn−1

un−1h,− v

n−1h,+ ds

=: bQn(unh, vnh)−

∫Σn−1

un−1h,− v

n−1h,+ ds.

For the local spaces V(n)h , n = 1, . . . , N , we now introduce the basis functions

V(n)h = spanϕnj Nn

j=1,

and we obtain from the discrete problem (2.22a) the linear system

Lhuh :=

A1

−B2 A2

. . . . . .

−BN AN

u1

u2...uN

=

f 1

f 2...fN

=: fh, (4.1)

with the matrices

An[i, j] := bQn(ϕnj , ϕni ) for i, j = 1, . . . , Nn

on the diagonal for n = 1, . . . , N , and the matrices

Bn[i, k] :=

∫Σn−1

ϕn−1k,− ϕ

n−1i,+ ds for k = 1, . . . , Nn−1 and i = 1, . . . , Nn.

on the lower off diagonal for n = 2, . . . , N . Moreover, the right hand sides are given by

fn[i] := lh(ϕni ), i = 1, . . . , Nn,

for n = 1, . . . , N . The linear system (4.1) can be solved by solving the local space-time problemssequentially from one space-time patch to the next space-time patch, i.e., like a time steppingscheme

Anun = fn + Bnun−1 for n = 2, . . . , N.

In this work, we will solve the linear system (4.1) by using a space-time multigrid approachsimilar to that one proposed in [19]. In particular, we use an (inexact) damped Jacobi scheme assmoother, i.e.,

uk+1h = ukh + ωD−1

h

[fh − Lhu

kh

]for k = 1, 2, . . . ,

where we use the block diagonal matrix Dh := diagAnNn=1 and the damping parameter ω =1

2,

see also [19]. We speed up the application of the smoothing iteration by replacing the exact inverseof Dh by some appropriate approximation. In detail, we will apply one iteration of an algebraicmultigrid solver (hypre [17, 16]) with respect to the diagonal matrices An, n = 1, . . . , N , i.e. foreach single space-time patch Qn. For the single patch case this type of solvers where successfullyused in [35]. For the space-time multigrid approach, we construct a space-time hierarchy by


always combining two space-time patches to one coarser space-time patch, where we always applystandard coarsening in time and space direction. We then have all the components available forsetting up a standard multigrid V-cycle. The advantage is that this method is fully parallel withrespect to space and time, since we use an additive smoother in time direction and apply standardparallel solvers in space direction. Moreover, we will use one iteration of this space-time multigridV-cycle as a preconditioner for the GMRES method.

If the IgA maps Φn : Q→ Qn, n = 1, . . . , N , preserve the tensor product structure of the IgAbasis functions ϕni , we can use this information to save assembling time and storage costs for thelinear system (4.1). In this case we can write the basis functions ϕni in the form

ϕni (x, t) = φnix(x)ψnit(t) with ix ∈ 1, . . . , Nn,x and it ∈ 1, . . . , Nn,t,

with Nn = Nn,xNn,t. Using this representation, we can write the matrices An, n = 1, . . . , N as

An = Mn,x ⊗Kn,t + Kn,x ⊗Mn,t,

with the standard mass and stiffness matrices with respect to space

Mn,x[ix, jx] :=

∫Ω

φnjxφnix dx, Kx[ix, jx] :=

∫Ω

∇xφnjx · ∇xφ

nix dx,

where ix, jx = 1, . . . , Nn,x and corresponding matrices with respect to time

Kn,t[it, jt] :=

∫ tn

tn−1

∂tψnjt(ψ

nit + θnhn∂tψ

nit) dt+ ψnjt(tn−1)ψnit(tn−1),

Mn,t[it, jt] :=

∫ tn

tn−1

ψnjt(ψnit + θnhn∂tψ

nit) dt,

with it, jt = 1, . . . , Nn,t. The matrices on the off diagonal Bn, n = 2, . . . , N , can be written in theform

Bn := Mn,x ⊗Nn,t,

with the matrices

Mn,x[ix, kx] :=

∫Ω

φn−1kx

φnix dx and Nn,t[it, kt] := ψn−1kt

(tn−1)ψnit(tn−1),

where ix = 1, . . . , Nn,x, kx = 1, . . . , Nn−1,x, it = 1, . . . , Nn,t and kt = 1, . . . , Nn−1,t.

5 Numerical examples

In the following, we present numerical examples supporting the theory developed in this paper.In Section 5.1, we verify the a-priori error estimate from Theorem 3.2 for higher order B-Splines.In Section 5.2, we show the parallel performance of the space-time solver given in Section 4.

5.1 Convergence studies

In this example, the problem is considered on the two dimensional space-time cylinder Q =Ω× (0, 4) with Ω = (0, 1). We choose homogeneous boundary conditions and the source function

f(x, t) = π sin(πx)(1

2cos(

π

2(t+ 1)) + π sin(

π

2(t+ 1))).


Hence, the exact solution is given by

u(x, t) = sin(πx) sin(π

2(t+ 1)). (5.1)

The space-time cylinder Q is decomposed into four space-time patches Qn = Ω× (tn−1, tn) wheret0, t1, t2, t3, t4 = 0, 1, 2, 3, 4, see Fig. 2(a). The problem has been solved on a sequence ofmeshes with h0, ..., hi, hi+1, ..., with hi = 2−i. According to Fig. 2(a), the mesh on Q1 and Q3

has one additional refinement. We discretize the problem using B-Splines of degree p = 2, 3, 4.Throughout all tests, θn is chosen to be 1 for all patches. The final linear system (4.1) is solved bymeans of a direct solver, where use used the PARDISO 5.0.0 Solver Project [46, 45]. The algorithmis realized in the isogeometric open source C++ library G+SMO [39]. The solution uh on a coarsemesh with h = 0.25 is visualized in Fig. 2(a). The error in the dG-norm and the convergencerates are presented in Table 1 and plotted in Fig. 2(b).

(a)

10-2 10-1 100

meshsize h

10-12

10-10

10-8

10-6

10-4

10-2

100

errorin

dG-norm

‖·‖dG

p=2

p=3

p=4

O(h2)

O(h3)

O(h4)

(b)

Fig. 2. (a) The solution uh on Q having non-matching meshes across the interface after two uniform refinements of theinitial mesh. (b) Convergence plots for polynomial degrees p = 2, 3, 4.

p = 2 p = 3 p = 4refinement error eoc error eoc error eoc

0 2.85633E-02 - 3.85617E-02 - 9.18731E-03 -1 5.68232E-02 2.33 7.15551E-03 2.43 7.87619E-04 3.542 1.34212E-02 2.08 8.11296E-04 3.14 4.62549E-05 4.093 3.30721E-03 2.02 9.84754E-05 3.04 2.90675E-06 3.994 8.23704E-04 2.01 1.22142E-05 3.01 1.84067E-07 3.985 2.05716E-04 2.00 1.52376E-06 3.00 1.16139E-08 3.996 5.14138E-05 2.00 1.90375E-07 3.00 7.29917E-10 3.997 1.28522E-05 2.00 2.37936E-08 3.00 4.85647E-11 3.91

Table 1. Error in the dG-norm and convergence rate for B-Spline degree 2, 3 and 4.


We observe that the obtained convergence rates coincide with the theoretically predicted ratesfrom Theorem 3.2 for smooth solutions u. To be more precise, we observe ‖u−uh‖dG behaves likeO(hp), where p is the B-Spline degree.

5.2 Parallel solver studies

Here we apply the parallel multigrid solver that was introduced in Section 4 to solve the arisinglinear systems for the case p = 1, i.e., for lowest order splines. In detail, we consider the simulationtime T = 1 and the computational domain Ω ⊂ R3 given by the control points

(0, 0, 0)>, (1, 0, 0)>, (1, 1, 0)>, (0, 1, 0)>, (−1

4,−1

4, 1)>, (1, 0, 1)>, (1, 1, 1)>, (−1

4,5

4, 1)>,

see also Figure 3.

Fig. 3. Computational spatial domain Ω decomposed into 4096 elements (left) and distributed over 32 processors (right).The numerical solution given in Table 2 is plotted at t = 0.5.

For the initial space-time, mesh we use one space-time patch (N = 1) which is decomposedinto 64 elements in space and 8 elements wrt time. We then apply uniform refinement wrt tospace, and, at the same time, we increase the number of space-time patches by a factor of two,i.e. uniform refinement in space and time. Throughout all computations we use the parameterθn = 0.2 for all space-time patches. Moreover, we assemble the linear systems and apply theparallel space-time multigrid solver, discussed in Section 4, as a preconditioner for the GMRESmethod. For the problem in space, we make use of the software library MFEM [40], where theAMG library hypre is used as parallel solver in space. For the time parallelization, we use thesoftware developed in [19]. For all examples, we stop the GMRES method until a relative residualerror of 10−12 is reached. In Table 2, we present the numerical results for the manufacturedsolution

u(x, t) = sin(πx1) sin(πx2) sin(πx3) sin(πt),

which is regular. For this example, we observe the optimal convergence rates in the dG-norm,which is predicted by the theory given in Theorem 3.2. For the L2(Q)-norm, we also obtain theoptimal rates. Furthermore, we also obtain quite small iteration numbers for the preconditionedGMRES method. In Table 2, we denote the number of cores which are used for the hypre AMGsolver and the number of cores which are used for the time parallelization by cx and ct, respectively.Hence, we use cxct cores overall. Finally, we can solve a linear system consisting of 9 777 365 568


N dof per patch overall dof ||u− uh||L2(Q) eoc ‖u− uh‖dG eoc cx ct cores iter time [s]

1 1 125 1 125 2.41829E-02 - 3.56223E-01 - 1 1 1 1 0.032 6 561 13 122 6.27531E-03 1.95 1.77477E-01 1.01 1 2 2 13 1.874 44 217 176 868 1.58802E-03 1.98 8.86255E-02 1.00 1 4 4 15 21.478 323 433 2 587 464 3.98310E-04 2.00 4.42868E-02 1.00 4 8 32 15 100.4816 2 471 625 39 546 000 9.96216E-05 2.00 2.21376E-02 1.00 32 16 512 17 94.3232 19 320 201 618 246 432 2.49008E-05 2.00 1.10675E-02 1.00 256 32 8192 17 162.9064 152 771 337 9 777 365 568 6.22393E-06 2.00 5.53340E-03 1.00 2048 64 131072 17 211.33

Table 2. Convergence results in the L2(Q)-norm and dG-norm for a regular solution as well as iteration numbers andsolving times for the parallel space-time multigrid preconditioned GMRES method.

unknowns in less than 4 minutes. We also observe a reasonable weak parallel efficiency of at least75%, which is mainly affected by the efficiency of the parallel AMG solver hypre.

In Table 3, we give the convergence rates for the manufactured solution

u(x, t) = sin(πx1) sin(πx2) sin(πx3)(1− t)α ∈ Hs,α+ 12−ε(Q)

for α = 0.75, for an arbitrary s ≥ 2 and for an arbitrary small ε > 0, which has lower regularitywrt time, see also [43]. By Theorem 3.2, the asymptotic convergence rate wrt h is then (almost)given by 0.75. In Table 3, we observe a convergence rate of one, since we are still in the pre-asymptotic range. For the L2(Q)-error, we obtain a reduced convergence rate, which will be 1.25in the asymptotic range. We also observe that the solver is not effected at all by the regularity ofthe solution.


1 1 125 1 125 1.96957E-02 - 3.15855E-01 - 1 1 1 1 0.032 6 561 13 122 5.06642E-03 1.96 1.58494E-01 0.99 1 2 2 13 1.864 44 217 176 868 1.27969E-03 1.99 7.92759E-02 1.00 1 4 4 15 21.468 323 433 2 587 464 3.24639E-04 1.98 3.96365E-02 1.00 4 8 32 15 100.4916 2 471 625 39 546 000 8.44577E-05 1.94 1.98187E-02 1.00 32 16 512 17 94.3532 19 320 201 618 246 432 2.33578E-05 1.85 9.91105E-03 1.00 256 32 8192 17 163.0164 152 771 337 9 777 365 568 7.20492E-06 1.70 4.95723E-03 1.00 2048 64 131072 17 211.47

Table 3. Convergence results in the L2(Q)-norm and dG-norm for a low regularity solution as well as iteration numbersand solving times for the parallel space-time multigrid preconditioned GMRES method.

In the next example, we use a different manufactured solution

u(x, t) = cos(βx1) cos(βx2) cos(βx3)(1− t)α ∈ Hs,α+ 12−ε(Q)

for α = 0.75 and β = 0.3, for an arbitrary s ≥ 2, and for an arbitrary small ε > 0, which has thesame regularity as the solution from the previous example. In Table 4, we observe the expectedconvergence rates predicted by Theorem 3.2.

All parallel computations have been performed on the cluster Vulcan BlueGene/Q at Liver-more, U.S.A.

6 Conclusions

We have presented and analyzed a time-multipatch discontinuous Galerkin space-time IgA methodfor solving initial-boundary value problems for linear parabolic partial differential equations. Themethod proposed uses discontinuous Galerkin techniques with time-upwind fluxes for establishingthe communication of the discrete solution across the time patch interfaces. Furthermore, time-upwind diffusion techniques was used for stabilizing the time discretization within each patch.



1 1 125 1 125 7.10407E-03 - 1.58022E-02 - 1 1 1 1 0.032 6 561 13 122 2.66063E-03 1.42 8.88627E-03 0.83 1 2 2 13 2.004 44 217 176 868 1.00177E-03 1.41 5.41668E-03 0.71 1 4 4 15 21.488 323 433 2 587 464 3.78976E-04 1.40 3.33881E-03 0.70 4 8 32 15 100.5716 2 471 625 39 546 000 1.44246E-04 1.39 2.05545E-03 0.70 32 16 512 17 94.4332 19 320 201 618 246 432 5.53509E-05 1.38 1.25859E-03 0.71 256 32 8192 17 171.8364 152 771 337 9 777 365 568 2.14541E-05 1.37 7.65921E-04 0.72 2048 64 131072 17 211.49

Table 4. Convergence results in the L2(Q)-norm and dG-norm for a low regularity solution as well as iteration numbersand solving times for the parallel space-time multigrid preconditioned GMRES method.

A complete discretization error analysis was developed in a suitable energy norm including thecase where the solution can exhibit different regularity behavior with respect the space andtime directions. The convergence rate estimates were confirmed by numerical experiments. Weproposed fast techniques for generating and solving the huge system of IgA equations on massivelyparallel computers. The parallel experiments were performed for a 3d spatial domain Ω yieldinga 4d space-time cylinder Q = Ω × (0, T ) but only for the case p = 1 where the IgA coincideswith the FEM. In this paper, we always assumed that the spatial computational domain Ω hasa singlepatch representation. The multipatch representation of Ω, which is more important inpractice, in connection with dG coupling in space [37] and Dual-Primal IsogEometric Tearingand Interconnection (IETI-DP) solution techniques [22, 21] is work in progress. This approach isquite flexible with respect to the adaption of the discretization to the behavior of the solution.At the same, it allows a fast generation and solution of the system of IgA equations due to thefact that the local space-time patches into which the space-time cylinder Q is decomposed havestill tensor product structure. A complete unstructured decomposition of the space-time cylinderQ into patches, which was considered in [42], loses this structure, and, therefore, an efficientimplementation is cumbersome.

Acknowledgments

This work was supported by the Austrian Science Fund (FWF) under the grants NFN S117-03and DK W1214-04.

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